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Applied Mechanics
Special Issues
Mechanics and Control using Fractional Calculus
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Mechanics and Control using Fractional Calculus

A special issue of Applied Mechanics (ISSN 2673-3161).

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 18577

Special Issue Editors

Prof. Dr. Timothy Sands

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Guest Editor
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
Interests: robotics; deterministic artificial intelligence; motion mechanics; system identification; guidance, navigation, and control; autonomy; technical translation; nuclear deterrence
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Kevin Bollino

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Guest Editor
Department of Electrical and Computer Engineering, George Mason University, Fairfax, VA 22030, USA
Interests: optimal control; autonomous vehicles; orbital mechanics
Special Issues, Collections and Topics in MDPI journals
Prof. Matt Crook

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Guest Editor
Space Systems Academic Group, Naval Postgraduate School, Monterey, CA 93943, USA
Interests: space systems; space system operations; MILSATCOM operations; satellite ground systems; CubeSats; CubeSat launch systems; national space systems; DoD space systems
Prof. Ross Eldred

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Guest Editor
Department of Systems Engineering, Naval Postgraduate School, Monterey, CA 93943, USA
Interests: UUVs; littoral environments/mine warfare; space systems; exploration of extreme environments; human–systems integration; cryogenics

Special Issue Information

Dear Colleague,

This Special Issue provides an advanced forum for studies related to fractals and fractional calculus and their applications in different fields of science and engineering focused on mechanics (electrical and motion) and control. The Issue publishes regular research papers, reviews, and even short notes. Our aim is to encourage authors to publish experimental and theoretical results in as much detail as possible; therefore, there is no restriction on the length of the papers as details of the experiment and any calculations must be provided so that the results can be reproduced. Manuscripts regarding research proposals and research ideas are particularly welcome. Electronic files containing details of calculations and experimental procedures can be deposited as supplementary material.

Topics include: fractional operators applied to innovative methods and algorithms, theoretical methods and applications, e.g., dynamic systems, bifurcation, and chaos; topology; mathematical physics; optimization and control theory; image analysis; classical mechanics; nonlinear problems; fractional and fractal signals; fractional and fractal dynamics in engineering mechanics; fractional and fractal dynamics in chemical systems; complex fractional dynamics in engineering.

Prof. Dr. Timothy Sands
Prof. Dr. Kevin Bollino
Prof. Matt Crook
Prof. Ross Eldred
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Applied Mechanics is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional calculus
  • electromechanics
  • motion mechanics
  • controls
  • fractals
  • chaos
  • neural systems
  • signal processing
  • mechatronics

Published Papers (7 papers)

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Research

51 pages, 1274 KiB  
Article
Quantum Mechanics and Control Using Fractional Calculus: A Study of the Shutter Problem for Fractional Quantum Fields
by Jonathan Blackledge
Appl. Mech. 2022, 3(2), 413-463; https://doi.org/10.3390/applmech3020026 - 12 Apr 2022
Cited by 3 | Viewed by 2026
Abstract
The ‘diffraction in space’ and the ‘diffraction in time’ phenomena are considered in regard to a continuously open, and a closed shutter that is opened at an instant in time, respectively. The purpose of this is to provide a background to the principal [...] Read more.
The ‘diffraction in space’ and the ‘diffraction in time’ phenomena are considered in regard to a continuously open, and a closed shutter that is opened at an instant in time, respectively. The purpose of this is to provide a background to the principal theme of this article, which is to extend the ‘quantum shutter problem’ for the case when the wave function is determined by the fundamental solution to a partial differential equation with a fractional derivative of space or of time. This involves the development of Green’s function solutions for the space- and time-fractional Schrödinger equation and the time-fractional Klein–Gordon equation (for the semi-relativistic case). In each case, the focus is on the development of primarily one-dimensional solutions, subject to an initial condition which controls the dynamical behaviour of the wave function. Coupled with variations in the fractional order of the fractional derivatives, illustrative example results are provided that are based on presenting space-time maps of the wave function; specifically, the probability density of the wave function. In this context, the paper provides a case study of fractional quantum mechanics and control using fractional calculus. Full article
(This article belongs to the Special Issue Mechanics and Control using Fractional Calculus)
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19 pages, 7712 KiB  
Article
On the Modeling of Ship Stiffened Panels Subjected to Uniform Pressure Loads
by Efstathios L. Platypodis and Konstantinos N. Anyfantis
Appl. Mech. 2022, 3(1), 125-143; https://doi.org/10.3390/applmech3010010 - 01 Feb 2022
Cited by 2 | Viewed by 4430
Abstract
Stiffened panels constitute structural assemblies of the entire ship hull, i.e., double bottom, side shell, deck areas, etc. Prescriptive dimensioning of the stiffeners (web thickness and height and flange thickness and breadth) is solely based on the application of beam bending theories. This [...] Read more.
Stiffened panels constitute structural assemblies of the entire ship hull, i.e., double bottom, side shell, deck areas, etc. Prescriptive dimensioning of the stiffeners (web thickness and height and flange thickness and breadth) is solely based on the application of beam bending theories. This work is divided into two parts. The first part involves the assessment of the structural response of one-way (single-bay) stiffened panels under uniform pressure. The objective is to evaluate the effectiveness of alternative approaches in obtaining accurate secondary stress fields. Both state-of-the-art analytical solutions (Paik, Schade, CSR, Miller) and numerical calculation tools (finite element analysis (FEA)) are employed and compared for this purpose. When it comes to cross-stiffened panels, numerical methods are usually used within the design process which is time demanding. The second part of this work focuses on the development of a fast, yet effective, prescriptive approach. This approach will allow the dimensioning of the longitudinal stiffeners by considering the secondary stress field. Combining finite element analysis and the Euler–Bernoulli bending theory, the effect of the transverse stiffeners to the longitudinal stiffeners is examined in order to estimate the type of support on the boundaries of the transverse stiffeners. Determining the type of support, will make it possible to apply the classical formula of bending stress instead of using finite element analysis, thus limiting the computational cost. Preliminary calculations show that most of the examined cases may be treated as fully clamped beams subjected to uniform pressure. Full article
(This article belongs to the Special Issue Mechanics and Control using Fractional Calculus)
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13 pages, 1131 KiB  
Article
Subspace Reduction for Stochastic Planar Elasticity
by Harri Hakula and Mikael Laaksonen
Appl. Mech. 2022, 3(1), 1-13; https://doi.org/10.3390/applmech3010001 - 22 Dec 2021
Viewed by 2114
Abstract
Stochastic eigenvalue problems are nonlinear and multiparametric. They require their own solution methods and remain one of the challenge problems in computational mechanics. For the simplest possible reference problems, the key is to have a cluster of at the low end of the [...] Read more.
Stochastic eigenvalue problems are nonlinear and multiparametric. They require their own solution methods and remain one of the challenge problems in computational mechanics. For the simplest possible reference problems, the key is to have a cluster of at the low end of the spectrum. If the inputs, domain or material, are perturbed, the cluster breaks and tracing of the eigenpairs become difficult due to possible crossing of the modes. In this paper we have shown that the eigenvalue crossing can occur within clusters not only by perturbations of the domain, but also of material parameters. What is new is that in this setting, the crossing can be controlled; that is, the effect of the perturbations can actually be predicted. Moreover, the basis of the subspace is shown to be a well-defined concept and can be used for instance in low-rank approximation of solutions of problems with static loading. In our industrial model problem, the reduction in solution times is significant. Full article
(This article belongs to the Special Issue Mechanics and Control using Fractional Calculus)
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20 pages, 8107 KiB  
Article
Identification of Fractional Damping Parameters in Structural Dynamics Using Polynomial Chaos Expansion
by Marcel S. Prem, Michael Klanner and Katrin Ellermann
Appl. Mech. 2021, 2(4), 956-975; https://doi.org/10.3390/applmech2040056 - 30 Nov 2021
Cited by 1 | Viewed by 2360
Abstract
In order to analyze the dynamics of a structural problem accurately, a precise model of the structure, including an appropriate material description, is required. An important step within the modeling process is the correct determination of the model input parameters, e.g., loading conditions [...] Read more.
In order to analyze the dynamics of a structural problem accurately, a precise model of the structure, including an appropriate material description, is required. An important step within the modeling process is the correct determination of the model input parameters, e.g., loading conditions or material parameters. An accurate description of the damping characteristics is a complicated task, since many different effects have to be considered. An efficient approach to model the material damping is the introduction of fractional derivatives in the constitutive relations of the material, since only a small number of parameters is required to represent the real damping behavior. In this paper, a novel method to determine the damping parameters of viscoelastic materials described by the so-called fractional Zener material model is proposed. The damping parameters are estimated by matching the Frequency Response Functions (FRF) of a virtual model, describing a beam-like structure, with experimental vibration data. Since this process is generally time-consuming, a surrogate modeling technique, named Polynomial Chaos Expansion (PCE), is combined with a semi-analytical computational technique, called the Numerical Assembly Technique (NAT), to reduce the computational cost. The presented approach is applied to an artificial material with well defined parameters to show the accuracy and efficiency of the method. Additionally, vibration measurements are used to estimate the damping parameters of an aluminium rotor with low material damping, which can also be described by the fractional damping model. Full article
(This article belongs to the Special Issue Mechanics and Control using Fractional Calculus)
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16 pages, 378 KiB  
Article
Morphology-Dependent Resonances in Two Concentric Spheres with Variable Refractive Index in the Outer Layer: Analytic Solutions
by Umaporn Nuntaplook and John A. Adam
Appl. Mech. 2021, 2(4), 781-796; https://doi.org/10.3390/applmech2040045 - 07 Oct 2021
Viewed by 2889
Abstract
In many applications constant or piecewise constant refractive index profiles are used to study the scattering of plane electromagnetic waves by a spherical object. When the structured media has variable refractive indices, this is more of a challenge. In this paper, we investigate [...] Read more.
In many applications constant or piecewise constant refractive index profiles are used to study the scattering of plane electromagnetic waves by a spherical object. When the structured media has variable refractive indices, this is more of a challenge. In this paper, we investigate the morphology dependent resonances for the scattering of electromagnetic waves from two concentric spheres when the outer shell has a variable refractive index. The resonance analysis is applied to the general solutions of the radial Debye potential for both transverse magnetic and transverse electric modes. Finally, the analytic conditions to determine the resonance locations for this system are derived in the closed form of both modes. Our numerical results are provided with discussion. Full article
(This article belongs to the Special Issue Mechanics and Control using Fractional Calculus)
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17 pages, 808 KiB  
Article
Existence of Incompressible Vortex-Class Phenomena and Variational Formulation of Raleigh–Plesset Cavitation Dynamics
by Terry Eleftherios Moschandreou and Keith Christian Afas
Appl. Mech. 2021, 2(3), 613-629; https://doi.org/10.3390/applmech2030035 - 29 Aug 2021
Cited by 1 | Viewed by 1573
Abstract
The following article extends a decomposition to the Navier–Stokes Equations (NSEs) demonstrated in earlier studies by corresponding author, in order to now demonstrate the existence of a vortex elliptical set inherent to the NSEs. These vortice elliptical sets are used to comment on [...] Read more.
The following article extends a decomposition to the Navier–Stokes Equations (NSEs) demonstrated in earlier studies by corresponding author, in order to now demonstrate the existence of a vortex elliptical set inherent to the NSEs. These vortice elliptical sets are used to comment on the existence of solutions relative to the NSEs and to identify a potential manner of investigation into the classical Millennial Problem encompassed in Fefferman’s presentation. The article also presents the utilization of a recently developed versatile variational framework by both authors in order to study a related fluid-mechanics phenomena, namely the Raleigh–Plesset equations, which are ultimately obtained from the NSEs. The article develops, for the first time, a Lagrangian density functional for a closed surface which when minimized produced the Raleigh–Plesset equations. The article then proceeds with the demonstration that the Raleigh–Plesset equations may be obtained from this energy functional and identifies the energy dissipation predicted by the proposed Lagrangian density. The importance of the novel Raleigh–Plesset functional in the greater scheme of fluid mechanics is commented upon. Full article
(This article belongs to the Special Issue Mechanics and Control using Fractional Calculus)
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10 pages, 578 KiB  
Article
Spherical Indentation of a Micropolar Solid: A Numerical Investigation Using the Local Point Interpolation–Boundary Element Method
by Gaël Pierson, M’Barek Taghite, Pierre Bravetti and Richard Kouitat Njiwa
Appl. Mech. 2021, 2(3), 581-590; https://doi.org/10.3390/applmech2030033 - 21 Aug 2021
Cited by 2 | Viewed by 1437
Abstract
The load-penetration curve in elastic nanoindentation of an elastic micropolar flat by a diamond spherical punch is analyzed. The presented results are obtained by a specifically developed numerical tool based on a judicious combination of the conventional boundary element method and strong form [...] Read more.
The load-penetration curve in elastic nanoindentation of an elastic micropolar flat by a diamond spherical punch is analyzed. The presented results are obtained by a specifically developed numerical tool based on a judicious combination of the conventional boundary element method and strong form local point interpolation method. The results show that the usual linear relationship between the material depression and the square of the radius of the contact area is also valid in this case of micropolar elastic material. It is also shown that the relation between the indentation stress (applied load over the contact surface) and the indentation strain (ratio of contact radius by the punch radius) is linear. The proportionality coefficient which is none other than the indentation stiffness varies with the coupling factor of the micropolar elastic medium. A relation between the indentation stiffness of a micropolar solid and that of a conventional solid with the same Young modulus and Poisson ratio is derived. Full article
(This article belongs to the Special Issue Mechanics and Control using Fractional Calculus)
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